Fuzzy Binary Relations Research Articles

Fuzzy Conditional Probability Relations and their Applications in Fuzzy Information Systems

In our real-world applications, data may be imprecise in which levels or degrees of preciseness of data are intuitively different. In this case, fuzzy set expressions are considered as an alternative to represent the imprecise data. In general, the degree of similarity relationship between two fuzzy (imprecise) data in real-world applications may not necessarily be symmetric or transitive. In order to provide such a degree of similarity between two fuzzy data, we introduced the fuzzy conditional probability relation. The concept of a fuzzy conditional probability relation may be considered as a concrete example of weak similarity relation which in turn is a special type of fuzzy binary relation generalizing similarity relation. Two important applications concerning the application of Knowledge Discovery and Data Mining (KDD) in the presence of a fuzzy data table (usually called fuzzy information system), namely removing redundant objects and recognizing partial or total dependency of (domain) attributes, are considered induced by the fuzzy conditional probability relation. Here, the fuzzy information system contains precise as well as imprecise data (fuzzy values) about objects of interest characterized by some attributes. Related to the dependency of attributes, we introduce the fuzzy functional dependency that satisfies Armstrong’s Axioms. In addition, we also discuss some interesting applications such as approximate data reduction and projection, approximate data querying and approximate joining in order to extend the query system.

Algorithms for computing the min-transitive closure and associated partition tree of a symmetric fuzzy relation

Based on a previously derived weight-driven algorithm for the computation of the T-transitive closure of an arbitrary binary fuzzy relation on a finite universe, with T a triangular norm, we establish three algorithms for the computation of the min-transitive closure of a symmetric matrix with elements in [0,1]. As a by-product, these algorithms enable to generate the partition tree associated with the min-transitive closure of the given matrix in descending order of the cutting parameter.

Fuzzy modeling of farmers' knowledge for land suitability classification

In a case study, we demonstrate fuzzy modeling of farmers' knowledge (FK) for agricultural land suitability classification using GIS. Capture of FK was through rapid rural participatory approach. The farmer respondents consider, in order of decreasing importance, cropping season, soil color, soil texture, soil depth and slope as factors of suitability of their land for certain crops. Multi-class fuzzy sets using S-membership functions were generated for soil texture, soil depth and slope because of correlation or equivalence between farmers' definitions and scientific classifications of such land characteristics. In contrast, binary fuzzy relations, which are also fuzzy sets, were generated for cropping season and soil color because farmers' perceptions of such land characteristics are intrinsically binary. Despite variations in individual farmers' perceptions of land suitability, 12 unique FK rules for classifying land suitability were defined by hierarchical grouping of such different perceptions based on decreasing importance of factors. The FK rules form inference engines in combining fuzzy factor maps using appropriate fuzzy operators to create agricultural land suitability maps. Suitability maps resulting from application of Fuzzy AND and Fuzzy OR operators were found consistent with the FK rules. The FK-based suitability maps indicate either agreement or conflict with a land resource development plan (LRDP) for the case study area. Results of the study indicate usefulness of fuzzy modeling in FK-based classification of agricultural land suitability, which could provide useful information for optimum land-use planning.

A general approach to fuzzy concepts

AbstractThe paper proposes a flexible way to build concepts within fuzzy logic and set theory. The framework is general enough to capture some important particular cases, with their own independent interpretations, like “antitone” or “isotone” concepts constructed from fuzzy binary relations, but also to allow the two universes (of objects and attributes) to be equipped each with its own truth structure. Perhaps the most important feature of our approach is that we do not commit ourselves to any kind of logical connector, covering thus the case of a possibly non‐commutative conjunction too. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cutlike semantics for fuzzy logic and its applications

Each fuzzy set can be represented by a nested system of ordinary sets—its a-cuts. There is an extensive literature on fuzzy sets devoted to problems of the following kind: is it possible to reduce operations with fuzzy sets to operations with their a-cuts? Is it possible to reduce properties of fuzzy relations to properties of their a-cuts? More generally, can a fuzzy concept be represented by a collection of corresponding crisp concepts? Klir and Yuan (1995) speak of cutworthiness. We attempt to provide a general solution to this problem. The way we proceed is thus: a structure for fuzzy predicate logic can be represented by a nested system of crisp structures. The system of crisp structures can be used to define semantics of fuzzy predicate logic in an alternative way by using the nested structure and Boolean connectives only. Answers to the above questions are then obtained by simple application of the obtained general results; we present some examples (extension principle, properties of binary fuzzy relations, fuzzy automata).

A new method for group decision support based on ELECTRE III methodology

Group decision is usually understood as the reduction of different individual preferences on a given set to a single collective preference. At present, there are few approaches which solve the group ranking problem with multiple criteria in a widely acceptable way. Often, they rest on a poor heuristic which makes a decision about consensus ranking difficult to support. This paper presents an extension of the ELECTRE III multicriteria outranking methodology to assist a group of decision makers with different value systems to achieve a consensus on a set of possible alternatives. Our proposal starts with N individual rankings and N corresponding valued preference functions, and uses the natural heuristic provided by ELECTRE methodology for obtaining a fuzzy binary relation representing the collective preference. A comparison of this method with PROMETHEE II for group decision is carried out. We found that, in this particular application, the proposed heuristic based on majority rules combined with concessions to significant minorities, performs relatively better than a compensatory scheme based on a net flow weighted sum function.

Using fuzzy sets to improve cluster labelling in unsupervised classification

This paper presents a fuzzy set approach to improving the cluster labelling process for unsupervised classification. Assuming normality for spectral data distribution, this approach represents both spectral clusters and the spectral signatures of information classes as fuzzy sets, with their fuzzy membership functions being defined using spectral means and variances. A fuzzy similarity index, defined through a fuzzy binary relation between each cluster fuzzy set and its corresponding class fuzzy set, is derived by applying a max-min fuzzy composition to the paired fuzzy sets. A second max-min fuzzy composition was performed to determine the final labelling of individual clusters based on these indices. Using a Thematic Mapper (TM) image of Imperial Valley, California as the test dataset, the approach was compared with both a human interpretation and a supervised classification resulting from the maximum-likelihood classifier. Both the process and the results indicated that the fuzzy approach surpassed the human interpretation in speed and equated the supervised classification in accuracy. In comparison to the Bayesian maximum-likelihood classifier, this approach supports a quick identification of mixed clusters and missing classes and incomplete classification. In addition, the fuzzy labelling approach provides a great potential for rapid a posteriori cluster validation.

DOMINATION OF AGGREGATION OPERATORS AND PRESERVATION OF TRANSITIVITY

Aggregation processes are fundamental in any discipline where the fusion of information is of vital interest. For aggregating binary fuzzy relations such as equivalence relations or fuzzy orderings, the question arises which aggregation operators preserve specific properties of the underlying relations, e.g. T-transitivity. It will be shown that preservation of T-transitivity is closely related to the domination of the applied aggregation operator over the corresponding t-norm T. Furthermore, basic properties for dominating aggregation operators, not only in the case of dominating some t-norm T, but dominating some arbitrary aggregation operator, will be presented. Domination of isomorphic t-norms and ordinal sums of t-norms will be treated. Special attention is paid to the four basic t-norms (minimum t-norm, product t-norm, Łukasiewicz t-norm, and the drastic product).

Characterization of fuzzy preference structures through Łukasiewicz triplets

In this paper we characterize the reconstruction of a fuzzy set from its subsets by means of Łukasiewicz triplets. This result allows us to introduce a new definition of fuzzy strict preference, which is also satisfied in the crisp framework. The usual definitions of fuzzy indifference and fuzzy incomparability together with this one enable to construct and to characterize fuzzy preference structures from a reflexive fuzzy binary relation.

Algorithms for the computation of T-transitive closures

We present two weight-driven algorithms for the computation of the T-transitive closure of a symmetric binary fuzzy relation on a finite universe X with cardinality n (or, equivalently, of a symmetric (n/spl times/n)-matrix with elements in [0, 1]), with T a triangular norm. The first algorithm is proven to be valid for any triangular norm T, whereas the second algorithm is shown to be valid when T is either the minimum operator or an Archimedean triangular norm. Furthermore, we investigate how these algorithms can be appropriately adapted to generate the T-transitive closure of nonsymmetric binary fuzzy relations (or matrices) as well.

Computing fuzzy associations for the analysis of biological literature.

The increase of information in biology makes it difficult for researchers in any field to keep current with the literature. The MEDLINE database of scientific abstracts can be quickly scanned using electronic mechanisms. Potentially interesting abstracts can be selected by matching words joined by Boolean operators. However this means of selecting documents is not optimal. Nonspecific queries have to be effected, resulting in large numbers of irrelevant abstracts that have to be manually scanned To facilitate this analysis, we have developed a system that compiles a summary of subjects and related documents on the results of a MEDLINE query. For this, we have applied a fuzzy binary relation formalism that deduces relations between words present in a set of abstracts preprocessed with a standard grammatical tagger. Those relations are used to derive ensembles of related words and their associated subsets of abstracts. The algorithm can be used publicly at http:// www.bork.embl-heidelberg.de/xplormed/.

One-to-one correspondences between [formula omitted]-partitions, [formula omitted]-equivalences and [formula omitted]-pseudometrics

The generalization of the concept of a classical (or crisp) partition to that of a fuzzy partition, requires, in our researches, the choice of a family of fuzzy subsets which preserves the conditions characterizing crisp partitions; however, the usual constraints for the union and intersection of subsets are very limiting and seem to be far from the “fuzzy spirit”; so that the union of subsets in the partitions is forced to be “close to” the universe of discourse and the intersection of any two subsets in the partition is forced to be “close to” the empty set. Furthermore, since the classical definition of partition is very related to the union and intersection, the generalization of this concept has to be applicable for any extension of these operations for fuzzy sets. Apart from this, we need to redefine the concept of fuzzy partition to establish a one-to-one correspondence between fuzzy partitions and some kind of fuzzy equivalence relations. The different definitions that we found in the literature did not exactly verify these requirements, although some of them were more appropriate than others. Finally, we have given a definition of fuzzy partition of any fuzzy subset of the universe, which is a generalization of crisp partitions and of most of the different definitions of fuzzy partitions, which preserves the classical ideas of covering and disjointness and such that it is appropriate in our researches (Classification Problems, Fuzzy Questionnaires, Fuzzy Information Theory). Once our fuzzy partitions ( ε-partitions) are defined, it is necessary to find the fuzzy binary relation which generates and characterizes the fuzzy partitions. This fuzzy relation has to be, by analogy with the crisp case, a fuzzy “equivalence” relation (two points in the same fuzzy subset of the partition will be “related”, and two points in different subsets will not). In relation with this “equivalence” relation, it is also possible to obtain a way to measure the “distance” between two points, by means of the degree of relation that there exists between them, and therefore, by means of the membership degree to a same subset in the fuzzy partition. The definitions of these concepts and the relationships between them are the purpose of this paper.

Aggregation of fuzzy preferences: Some rules of the mean

This paper studies by means of reciprocal fuzzy binary relations the aggregation of preferences when individuals show their preferences gradually. We have characterized neutral aggregation rules through functions from powers of the unit interval in the unit interval. Furthermore, we have determined the neutral aggregation rules that are decomposable and anonymous. In this class of rules, the collective intensity of preference is the arithmetic mean of the values assigned by a function to the individual intensities of preference. We have also considered the neutral aggregation rules based on quasiarithmetic means. We have established that this class of rules generalizes the simple majority, when individuals have ordinary preferences and collective preferences are reciprocal.

Properties of the fuzzy composition based on aggregation operators

The main aim of this paper is to present the binary compositions based on aggregation operators as a generalization of sup- t and inf- s compositions, with t a t-norm and s a t-conorm. The composition of two fuzzy sets, two fuzzy binary relations and a fuzzy relation with a fuzzy set are analyzed to prove that their properties are analogous to the sup- t and inf- s composition properties. For this proposal, the aggregation which includes the usual conditions of this kind of operators is chosen.

Fuzzy Galois Connections

AbstractThe concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one‐to‐one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided.

On the construction of fuzzy preference structures

In this paper, we analyse the generalization of the classical method of the construction of a preference structure from a reflexive binary relation to the case of fuzzy binary relations. According to our approach, there are two interesting fuzzy preference structures we can construct from a given reflexive fuzzy binary relation. These two fuzzy preference structures correspond to the two extreme solutions of the system of functional equations in the method of Fodor and Roubens. We also prove that only one of two fuzzy preference structures allows its fuzzy relation of strict preference to be transitive with respect to the φ-transform of the Lukasiewicz t-norm when the reflexive fuzzy relation it is constructed from is also transitive with respect to the same t-norm. © 1998 John Wiley & Sons, Ltd.

Characterizable fuzzy preference structures

In this paper, we study the existence, construction and reconstruction of fuzzy preferencestructures. Starting from the definition of a classical preference structure, we propose anatural definition of a fuzzy preference structure, merely requiring the fuzzification of theset operations involved. Upon evaluating the existence of these structures, we discover thatthe idea of fuzzy preferences is best captured when fuzzy preference structures are definedusing a L U ukasiewicz triplet. We then proceed to investigate the role of the completenesscondition in these structures. This rather extensive investigation leads to the proposal of astrongest completeness condition, and results in the definition of a one-parameter class offuzzy preference structures. Invoking earlier results by Fodor and Roubens, the constructionof these structures from a reflexive binary fuzzy relation is then easily obtained. Thereconstruction of such a structure from its fuzzy large preference relation inevitable toobtain a full characterization of these structures in analogy to the classical case is morecumbersome. The main result of this paper is the discovery of a non-trivial characterizingcondition that enables us to fully characterize the members of a two-parameter class offuzzy preference structures in terms of their fuzzy large preference relation. As a remarkableside-result, we discover three limit classes of characterizable fuzzy preference structures,traces of which are found throughout the preference modelling literature.

Formula omitted]-partitions of the real line generated by idempotent shapes

The idea of generating fuzzy numbers as equivalence classes of particular T -equivalences on the real line R is fully exploited. Scales (or generators) are used to define certain (pseudo-)metrics on R . By means of a shape (function), these (pseudo-)metrics are then transformed into binary fuzzy relations on R . Shapes leading to T -equivalences, and hence to a class of fuzzy numbers forming a T -partition of R , are completely characterized in the case of a continuous generator. This characterization problem is shown to be closely related to determining the idempotents w.r.t. the T -addition of fuzzy numbers.

Equivalence and compositions of fuzzy rationality measures

An axiomatic basis for fuzzy rationality measures has already been introduced by the authors in a previous paper [5], formalizing the fact that there exist degrees of consistency when preferences over a fixed set of alternatives are expressed in terms of fuzzy binary preference relations. This paper deals with some practical consequences. On the one hand, similarities and compositions of fuzzy rationality measures are considered, showing natural ways of deriving new measures; on the other, if basic stability properties are introduced in order to assure that small intensity measurement errors never lead to big changes in the associate rationality value, it is shown that crisp (i.e., binary) rationality measures present serious difficulties when applied to fuzzy preference relations.

Fuzzy preference structures without incomparability

In this paper, we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability. This study is carried out for the fuzzy preference structures introduced recently by De Baets, Van de Walle and Kerre. A set of remarkable theorems gives detailed insight in the relationships between the sup- T transitivity of the fuzzy preference and indifference relations and the sup- T transitivity of the fuzzy large preference relation. Several paths of thought, involving t-norms with or without zero-divisors, are explored and, where required, illustrative counterexamples confirm the falsity of certain implications. Finally, we introduce the ( T , N )-Ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this Ferrers property.